Finite Group Databasebeta

← #12157287

Subgroups — #12157287

Order 1280.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2344C211yesmaximal, characteristic
2#10605225C211yesmaximal, characteristic
3#10602351C211yesmaximal, characteristic
4#10602347C211yesmaximal, characteristic
5#10600415C211yesmaximal, characteristic, complemented
6#10600405C211yesmaximal, characteristic, complemented
7#10600403C211yesmaximal, characteristic, complemented
8#10600399C211yesmaximal, characteristic, complemented
9#10600306C211yesmaximal, characteristic
10#10600299C211yesmaximal, characteristic
11#10600284C211yesmaximal, characteristic, complemented
12#10600260C211yesmaximal, characteristic, complemented
13#10600254C211yesmaximal, characteristic, complemented
14#10600251C211yesmaximal, characteristic, complemented
15#10600243C211yesmaximal, characteristic
16C2×C5×#117—21no—
17C22×C5×C4⋊C4C2211yescharacteristic
18C2×#4072C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C2×#4016C2211yescharacteristic
21C5×#389C2211yescharacteristic
22C5×#389C2211yescharacteristic
23C5×#387—21no—
24C5×#387—21no—
25C5×#381—21no—
26C5×#380C2211yescharacteristic
27C4×C5×C4⋊C4C2211yescharacteristic
28C4×C5×C4⋊C4C2211yescharacteristic
29C4×C5×C22⋊C4C2211yescharacteristic
30#65501C2211yescharacteristic
31C2×C22⋊#168C2211yescharacteristic
32C2×#3940C2211yescharacteristic
33C2×#3940C2211yescharacteristic
34C2×#3939C2211yescharacteristic
35C2×#3939C2211yescharacteristic
36#65238—21no—
37#65237—21no—
38#65224C2211yescharacteristic
39#65217C2211yescharacteristic
40#65217C2211yescharacteristic
41C2×C4⋊#168C2211yescharacteristic
42C2×C4⋊#168C2211yescharacteristic
43#64943—21no—
44#64936—21no—
45#64749C2211yescharacteristic
46#64749C2211yescharacteristic
47#64749C2211yescharacteristic
48#64749C2211yescharacteristic
49#64749C2211yescharacteristic
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#26014—51nomaximal, Sylow
60C23×C4×C5—21no—
61C23×C4×C5C2311yescharacteristic
62C5×#117—44no—
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4Q814yes—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4D411yescharacteristic
74C2×C5×C22⋊C4D411yescharacteristic
75C2×C42×C5—21no—
76C2×C42×C5—21no—
77C2×C42×C5C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C22⋊Dic5—21no—
80C2×C22⋊Dic5—21no—
81#4072—22no—
82#4072—41no—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89C2×#683—21no—
90C2×#683—21no—
91C2×#683—21no—
92C2×#683—21no—
93C2×C4×Dic5—21no—
94C2×#681—22no—
95C2×#681—22no—
96C2×#681—41no—
97C22×#168C2311yescharacteristic
98C22×#168C2311yescharacteristic
99C22×#168C2311yescharacteristic
100C22×#168C2311yescharacteristic
101#4016—22no—
102#4016—41no—
103C5×#95—21no—
104C5×#95—21no—
105C5×#95—21no—
106C5×#95—21no—
107C5×#95—21no—
108C5×#95D411yescharacteristic
109C5×#95D411yescharacteristic
110#3963—21no—
111#3963—21no—
112#3963—21no—
113C22⋊#168—22no—
114#3940—41no—
115#3940—41no—
116#3939—22no—
117#3939—22no—
118C4⋊#168—22no—
119C4⋊#168—41no—
120#2441—51no—
121#2344D511yescharacteristic
122#2100—51no—
123#2088—51no—
124#2086—51no—
125#2059—51no—
126#1989—51no—
127#1948—51no—
128#1921—51no—
129#1918—51no—
130#1854—51no—
131#1851—51no—
132#1824—51no—
133#1809—51no—
134#1736—51no—
135C24×C5C2×D411yescharacteristic
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—24no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—42no—
146C22×C4×C5—42no—
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×Q812yes—
153C22×C4×C5#4112yes—
154C22×C4×C5C2411yescharacteristic
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4—42no—
167C5×C4⋊C4—42no—
168C5×C4⋊C4C2×Q812yes—
169C5×C4⋊C4C2×Q812yes—
170C5×C22⋊C4—22no—
171C5×C22⋊C4—41no—
172C5×C22⋊C4—42no—
173C42×C5—21no—
174C42×C5—21no—
175C42×C5—21no—
176C42×C5—21no—
177C42×C5—42no—
178C42×C5—42no—
179C22⋊Dic5—42no—
180C22⋊Dic5—81no—
181C4⋊Dic5—21no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—42no—
189C4⋊Dic5—81no—
190C4⋊Dic5#4112yes—
191#683—42no—
192#683—42no—
193#683—42no—
194#683—81no—
195C4×Dic5—42no—
196#681—44no—
197C2×#168—21no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168—21no—
207C2×#168—21no—
208C2×#168#4112yes—
209C22×M16—51no—
210C2×#125—51no—
211C2×#117—21no—
212C2×#117—51no—
213C22×C4⋊C4C2×D511yescharacteristic
214C2×#107—51no—
215C2×#107—51no—
216C2×#106—51no—
217C2×#106—51no—
218C2×C4⋊C8—51no—
219C2×C4⋊C8—51no—
220C2×C22⋊C8—51no—
221#399—101no—
222#396—101no—
223#389—51no—
224#389—51no—
225#389—101no—
226#389C2×D511yescharacteristic
227#389C2×D511yescharacteristic
228#387—21no—
229#387—21no—
230#387—101no—
231#384—51no—
232#381—21no—
233#380—51no—
234#380C2×D511yescharacteristic
235C4×C4⋊C4C2×D511yescharacteristic
236C4×C4⋊C4C2×D511yescharacteristic
237C4×C22⋊C4C2×D511yescharacteristic
238#340—51no—
239#340—51no—
240#340—51no—
241#340—51no—
242#340—51no—
243#340—51no—
244#340—51no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#340—51no—
250#336—51no—
251#336—51no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5C22⋊Q812yes—
259C23×C5C22×D411yescharacteristic
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—48no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5C22⋊Q812yes—
291C2×C4×C5AGL(1,ℤ/8)12yes—
292C2×C4×C5C22×Q811yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5C2×#4111yescharacteristic
295C2×C4×C5C2×#4111yescharacteristic
296C2×C4×C5H(5,𝔽2)11yescharacteristic
297C2×C4×C52−1+411yescharacteristic
298C2×Dic5—22no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310#168—22no—
311#168—41no—
312#168—41no—
313#168—41no—
314C23×C4—21no—
315C23×C4C22×D511yescharacteristic
316C2×M16—102no—
317C2×M16—102no—
318C2×M16—201no—
319C22×C8—51no—
320C22×C8—51no—
321C22×C8—51no—
322C22×C8—51no—
323#125—102no—
324#125—201no—
325#117—44no—
326#117—102no—
327#117—201no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—101no—
337C2×C4⋊C4—101no—
338C2×C4⋊C4—101no—
339C2×C4⋊C4—101no—
340C2×C4⋊C4—101no—
341C2×C4⋊C4—101no—
342C2×C4⋊C4Dic1014yes—
343C2×C4⋊C4C22×D511yescharacteristic
344C2×C4⋊C4C22×D511yescharacteristic
345C2×C4⋊C4C22×D511yescharacteristic
346C2×C4⋊C4C22×D511yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—101no—
349C2×C22⋊C4—101no—
350C2×C22⋊C4C22⋊D511yescharacteristic
351C2×C22⋊C4C22⋊D511yescharacteristic
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—101no—
355C2×C42C22×D511yescharacteristic
356C2×C42C22×D511yescharacteristic
357#107—102no—
358#107—102no—
359#106—201no—
360#106—201no—
361C4⋊C8—102no—
362C4⋊C8—201no—
363C22⋊C8—102no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—101no—
370#95—101no—
371#95—101no—
372#95C22⋊D511yescharacteristic
373#95C22⋊D511yescharacteristic
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—41no—
378C22×C5—41no—
379C22×C5—41no—
380C22×C5—41no—
381C22×C5C2×C22⋊Q811yescharacteristic
382C22×C5#54611yescharacteristic
383C22×C5#54711yescharacteristic
384C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C5#57511yescharacteristic
386C22×C5#57711yescharacteristic
387C22×C5#57811yescharacteristic
388C4×C5—22no—
389C4×C5—22no—
390C4×C5—41no—
391C4×C5—42no—
392C4×C5—42no—
393C4×C5—41no—
394C4×C5—41no—
395C4×C5—81no—
396C4×C5—81no—
397C4×C5#53311yescharacteristic
398C4×C5#54411yescharacteristic
399C4×C5#55111yescharacteristic
400C4×C5#55211yescharacteristic
401Dic5—42no—
402Dic5—81no—
403Dic5—81no—
404Dic5—81no—
405C24C2×C22⋊D511yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—101no—
418C22×C4—101no—
419C22×C4—101no—
420C22×C4—101no—
421C22×C4C2×Dic1012yes—
422C22×C4#70912yes—
423C22×C4C2×C22⋊D511yescharacteristic
424C22×C4C2×C22⋊D511yescharacteristic
425C22×C4C2×C22⋊D511yescharacteristic
426C22×C4C2×C22⋊D511yescharacteristic
427C22×C4C2×C22⋊D511yescharacteristic
428C22×C4C23×D511yescharacteristic
429M16—204no—
430C2×C8—52no—
431C2×C8—101no—
432C2×C8—101no—
433C2×C8—101no—
434C2×C8—101no—
435C2×C8—101no—
436C2×C8—101no—
437C2×C8—101no—
438C2×C8—101no—
439C2×C8—101no—
440C2×C8—101no—
441C2×C8—101no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—52no—
452C4⋊C4—101no—
453C4⋊C4—101no—
454C4⋊C4—101no—
455C4⋊C4—101no—
456C4⋊C4—101no—
457C4⋊C4—101no—
458C4⋊C4—101no—
459C4⋊C4—202no—
460C4⋊C4—202no—
461C4⋊C4—202no—
462C4⋊C4—202no—
463C4⋊C4—401no—
464C4⋊C4—401no—
465C4⋊C4C2×Dic1012yes—
466C4⋊C4C2×Dic1012yes—
467C22⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—202no—
471C22⋊C4—401no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—42no—
477C42—42no—
478C42—202no—
479C2×C5—42no—
480C2×C5#243811yescharacteristic
481C2×C5#307711yescharacteristic
482C2×C5#308211yescharacteristic
483C2×C5#331711yescharacteristic
484C2×C5#333011yescharacteristic
485C2×C5#334511yescharacteristic
486C2×C5#335211yescharacteristic
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23#407012yes—
494C23C22×C22⋊D511yescharacteristic
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—48no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—102no—
526C2×C4—201no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4—201no—
537C2×C4#407012yes—
538C2×C4#407812yes—
539C2×C4C22×Dic1011yescharacteristic
540C2×C4C2×#70911yescharacteristic
541C2×C4D5×#4111yescharacteristic
542C2×C4D5×#4111yescharacteristic
543C2×C4#414911yescharacteristic
544C2×C4#415011yescharacteristic
545C8—102no—
546C8—201no—
547C8—201no—
548C8—201no—
549C5#2601411yescharacteristic, complemented, Sylow
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#407011yescharacteristic
558C22C2×#407811yescharacteristic
559C22#6615611yescharacteristic
560C22#6615811yescharacteristic
561C22#6616311yescharacteristic
562C22#6617111yescharacteristic
563C22#6617211yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—202no—
574C4—401no—
575C4—401no—
576C4—401no—
577C4#6585111yescharacteristic
578C4#6585311yescharacteristic
579C4#6586111yescharacteristic
580C4#6586511yescharacteristic
581C2—42no—
582C2#1061397811yescharacteristic
583C2#1061402711yescharacteristic
584C2#1061428111yescharacteristic
585C2#1061525011yescharacteristic
586C2#1061525811yescharacteristic
587C2#1061532811yescharacteristic
588C2#1061533511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2344C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenter#2344D5
hypocenterC5#26014
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #10600415C2semidirect
6. #10600405C2semidirect
7. #10600403C2semidirect
8. #10600399C2semidirect
11. #10600284C2semidirect
12. #10600260C2semidirect
13. #10600254C2semidirect
14. #10600251C2semidirect
549. C5#26014semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×#23442
#106052252
#106023512
#106023472
#106004152
#106004052
#106004032
#106003992
#106003062
#106002992
#106002842
#106002602
#106002542
#106002512
#106002432
#260145